A. Field of the Invention
The present invention relates to a method for the representation of physical phenomena extending in a bi- or tridimensional spatial domain through a semistructured calculation grid.
B. Description of the Related Art
The representation of physical phenomena occurs by computerized numerical simulation of the spatial domain to be discretised. This operation, known as spatial discretisation, divides the spatial domain in a plurality of simple sub-domains to cover it, more or less accurately, throughout its extension.
Therefore, an approximation of a continuous domain composed of infinite points is obtained by a discontinuous domain composed of a finite number of elements, called cells.
In a two-dimensional space, the domain extends on a surface and the cells are polygonally shaped, whose sides, usually straight-lined, connect a set of vertices, also called nodes. In a three-dimensional space, the domain extends on a volume and the cells are polyhedrally shaped, delimited by faces. Each face is polygonally shaped, whose sides form the edges of the cell, connect a set of vertices.
The organized set of elements of different typologies (cells, vertices, sides, faces and edges) discretising the space of the calculation domain is a calculation grid. Each one of the elements forming a calculation grid is detected, within its typology, by a single global index. The various global indices relative to a certain typology are stored in vectors.
The numerical description of a calculation grid indicates for each vertex, the coordinates corresponding to its spatial position with respect to a predetermined reference system and the connectivity with the adjacent vertices, i.e., the way the vertices are connected therebetween through the sides of the polygonal cells (in the two-dimensional space) or through the faces of the polyhedral cells (in the three-dimensional space). The grids can be of different types according to their connective peculiarities.
A calculation grid is structured when its vertices are connected therebetween according to a same scheme throughout the calculation domain. In this case, the connectivity of the vertices is governed by the law with which they are ordered, i.e., it allows the determination of the global indices of the vertices adjacent therebetween.
A calculation grid is nonstructured when there is no ordering law among the vertices. In this case, it is necessary to indicate explicitly the connections of each vertex with the adjacent ones.
However, the use of each one of these types of grids causes practical problems, addressed by the development of intermediate typologies of grids, i.e., grids combined therebetween in ways that were different every time. However, these attempts failed to solve the problem.
With reference to FIG. 1, the representations of a simple bidimensional domain, such as the one formed by a quarter of a circle, will be shown.
The grids A and B are structured grids and are the most commonly used. The cells, usually quadrangular (hexahedral in the three-dimensional space), forming them and dividing the physical space, are in a biunivocal relation with the square cells (cubical in the three-dimensional space) obtained dividing, by two perpendicular and numerable sheaves of straight lines (three sheaves of planes in the three-dimensional space), a rectangle (a parallelepiped in the three-dimensional space) in the indicial space, called block. Therefore, the cells of the physical space are ordered by layers and are easy to localize considering the pair (the triad in case of three-dimensional space) of the directional indices i, j relative to the layers the cell belongs to. Furthermore, all the elements of the grid (cells, vertices, sides, faces, edges) are in biunivocal correspondence therebetween.
A peculiarity of the structured grids is demonstrated by the possibility of defining a global index ij starting from the directional indices i, j by an algebraic relation, in order to trace back the directional indices starting from the global one. As an example of the relation between the directional indices and the global index of a structured grid, reference can be made to the cell identified with an asterisk in the grid A. This cell is detected by the pair of directional indices (i,j) corresponding to (3,2). For example, its global index ij can be expressed by the following relation EQU ij=NI*(j-1)+i=5*(2-1)+3=8
wherein NI is the total number of layers of vertices in the direction i, e.g., 5. From said relation it is possible to notice how vertices adjacent therebetween in the direction i (direction in the radial or horizontal generating line) differ by a unit in the value of their global index, where vertices adjacent therebetween in the direction j (direction of the azimuthal or vertical generating line) differ by a value NI in the value of their global index. PA1 1) besides the matrix of the physical coordinates of the vertices, whose data are casually recorded not being possible, in this case, to order them according to an index ordered by layers, it is necessary to define a connectivity matrix (CM) indicating for each cell the 3 nodes (4 in the three-dimensional space) forming the vertices thereof; PA1 2) localizing each physical element (cell or vertex) in the grid cannot be easily obtained through the relative structural indices and requires the use of maps on which they are numbered one by one; PA1 3) it is difficult to use numerical schemes of an order higher than the first, i.e., taking into consideration together with the values of the quantities in the reference cells and in those immediately adjacent to them, also the values of the quantities in the cells adjacent to the latter; PA1 4) identifying afterwards, e.g., after the construction of the grid, subdomains that can be given some characteristics, is complicated, if not impossible: for example, to delimit a wall portion to which a determined temperature is to be imposed and in absence of a specific law for ordering the vertices, it would be necessary to list all the indices of all the cells comprised therein; PA1 5) similar or even more complicated problems are evident when the results have to be visualized: the visualization of the structured grids is performed following the layers identified by the indices, while for the nonstructured grids it can occur only for analytically defined physical surfaces, e.g., planes. This requires a complicated procedure which includes the individuation of a visualizing grating on the chosen surface and the determination in the vertices of the values of the quantity to be visualized by interpolation with the values of the grid cells closest to said vertices; PA1 6) finally, the nonstructured grids based on triangular elements require often a greater number of cells with respect to the structured ones, as, while for the quadrangular cells of the structured grids considerable elongations are permitted and therefore they can be thickened next to the walls only in a direction perpendicular to them, not always the sides of the triangular cells can be so different therebetween, so that a thickening in a direction can involve a similar thickening in the orthogonal directions. In the three-dimensional space, such a thickening involves a cubic increase in the number of the cells. PA1 1) detecting one or more sub-blocks, each sub-block being formed by the set of the vertices of the grid comprised, for each one of the two (or three) indicial directions, between a pair of layers of vertices evolving in said indicial direction; PA1 2) fixing for each sub-block an indicial direction for grouping the vertices, called semistructuring direction, said direction detecting a sheaf of sub-meshes in said sub-block, each sub-mesh of said sheaf consisting in a number of vertices lined along the semistructuring direction in the same number of distinct initial positions; PA1 3) predetermining for each sub-block a number of distinct final positions lower than the number of distinct initial positions, in which the vertices for each sub-mesh are to be grouped; and PA1 4) grouping for each sub-block the vertices of each sub-mesh in said number of different final positions, as to obtain the same number of different vertices for each sub-mesh of the sub-block, detecting at the same time, for each sub-block, a set of virtual vertices corresponding to the difference between the initial and the final number of vertices of that sub-block.
The structured grid A, hereby called cylindrical structured grid, is the most natural and easiest to construct, but has the drawback of an excessive thickening of the meshes towards the center. This drawback, besides an evident waste of cells and therefore of memory, involves an inevitable thinning of the dimension of the cells. In certain cases, this could considerably increase the calculation time because of the need of drastically reducing the time step to respect, for example, the stability criterion of Courant.
The structured grid B, hereby indicated as Cartesian structured grid, does not involve the above mentioned drawback, but its construction is more complicated and in particular it shows a strong nonhomogeneity along the direction j with excessively deformed cells such as those placed along the diagonal of the indicial space. The nonhomogeneity makes the creation of a mathematical model extremely difficult.
The structured grid C, also known as multiblock structured grid, tries to solve all the above mentioned problems. The grid is formed by more blocks (rectangular in two dimensions or parallelepiped in three dimensions) differently connected. However, its modelization creates new problems. First, the localization of the cells requires the use of an additional index, besides the directional ones, to identify the single block taken into consideration, second, more complicated calculation schemes are necessary. This modelization requires in each direction, the knowledge of the values in the cells upstream and downstream of the current cell.
The grid D is nonstructured, composed of triangular cells (tetrahedral in the three-dimensional space). An advantage given by the use of nonstructured grids is that of being able to control at will the thickening of the cells in the domain without worrying about the limits imposed by the structure ordered by layers of cells, as it is necessary in a structured grid. However, the nonstructured grid has several drawbacks caused essentially by the lack of a natural correspondence with an indicial space. Among the drawbacks, the following can be considered:
This last drawback can be overcome using hybrids grids E, also represented in FIG. 1, wherein the domain is divided in two types of regions, depending if the discretisation in the cells therein is performed by nonstructured grids or structured grids. The latter are as a rule used in regions adjacent to the walls. These grids, proposed to overcome the drawbacks shown singularly by both the types of grids composing them and that have also been adopted in commercial codes, actually gather all the problems explained up to now. In fact, they require numerical solvers able to manage the structural peculiarities of both the grids without solving, rather worsening, the problems of the localization of the cells and of portions of the calculation domain as well as of data visualization.
Accordingly, there is a need to improve simulation of structuring grids in different industrial and technological areas, e.g. all those wherein a numerical simulation of a physical phenomenon extending in the space can be performed. Furthermore, there is a need to reduce memory usage and reduce calculation time for creating a structured grid.